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Relaxed locally correctable codes with nearly-linear block length and constant query complexity
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Chiesa, Alessandro, Gur, Tom and Shinkar, Igor (2021) Relaxed locally correctable codes with nearly-linear block length and constant query complexity. SIAM Journal of Computing . pp. 1395-1411. doi:10.1137/1.9781611975994.84 ISSN 0097-5397.
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Official URL: https://doi.org/10.1137/1.9781611975994.84
Abstract
Locally correctable codes (LCCs) are codes C: Σk → Σn which admit local algorithms that can correct any individual symbol of a corrupted codeword via a minuscule number of queries. One of the central problems in algorithmic coding theory is to construct O(1)-query LCC with minimal block length. Alas, state-of-the-art of such codes requires exponential block length to admit O(1)-query algorithms for local correction, despite much attention during the last two decades.
This lack of progress prompted the study of relaxed LCCs, which allow the correction algorithm to abort (but not err) on small fraction of the locations. This relaxation turned out to allow constant-query correction algorithms for codes with polynomial block length. Specifically, prior work showed that there exist O(1)-query relaxed LCCs that achieve nearly-quartic block length n = k4+α, for an arbitrarily small constant α > 0.
We construct an O(1)-query relaxed LCC with nearly-linear block length n = k1+α, for an arbitrarily small constant α > 0. This significantly narrows the gap between the lower bound which states that there are no O(1)-query relaxed LCCs with block length n = k1+o(1). In particular, this resolves an open problem raised by Gur, Ramnarayan, and Rothblum (ITCS 2018).
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||||
| Library of Congress Subject Headings (LCSH): | Coding theory, Computer algorithms, Computational complexity | ||||||
| Journal or Publication Title: | SIAM Journal of Computing | ||||||
| Publisher: | SIAM | ||||||
| ISBN: | 978-1-61197-599-4 | ||||||
| ISSN: | 0097-5397 | ||||||
| Official Date: | 2021 | ||||||
| Dates: |
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| Page Range: | pp. 1395-1411 | ||||||
| DOI: | 10.1137/1.9781611975994.84 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Reuse Statement (publisher, data, author rights): | “First Published in SIAM Journal of Computing in [volume and number, or year], published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.”) | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 11 June 2021 | ||||||
| Date of first compliant Open Access: | 14 June 2021 | ||||||
| RIOXX Funder/Project Grant: |
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